
where \(\delta _n \mathbf {E}s_n(z) -\bar{b}_n\)  obeys \(|\delta _n| \lesssim 1/(nv) \le 1 /\sqrt{n \beta _n}\)  (note that \(\bar{s}_n(z)\)  is a solution of the related equation \(s = (1-z-\beta _n s)^{-1})\) .
By (3.3.8) of [1]}, (REF ) has a unique solution satisfying the requirement that \(\mathbf {E}s_n(u+iv) \rightarrow 0\)  as \(v \rightarrow \infty \) :
\( \phantom{\,.} \mathbf {E}s_n(z) = \frac{1}{2 \beta _n z} \big (1 - z - \beta _n + \beta _n z \delta _n + \sqrt{(z + \beta _n -1 + \beta _n z \delta _n)^2 - 4 \beta _n z}\big ) \,.\) 
